Immunity and Simplicity for Exact Counting and Other Counting Classes

Institut für Informatik, Friedrich-Schiller-Universität Jena, 07740 Jena, Germany; rothe@informatik.uni-jena.de.

Received: January 1998
Accepted: March 1999

Abstract

Ko [26] and Bruschi [11] independently showed that, in some relativized world, PSPACE (in fact, ⊕P) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of relativized separations with immunity for PH and the counting classes PP, , and ⊕P in all possible pairwise combinations. Our main result is that there is an oracle A relative to which contains a set that is immune BPP^⊕P. In particular, this set is immune to PH^A and to ⊕P^A. Strengthening results of Torán [48] and Green [18], we also show that, in suitable relativizations, NP contains a -immune set, and ⊕P contains a PP^PH-immune set. This implies the existence of a -simple set for some oracle B, which extends results of Balcázar et al. [2,4]. Our proof technique requires a circuit lower bound for “exact counting” that is derived from Razborov's [35] circuit lower bound for majority.